Multidimensional modulation (MD modulation) is modifying or multiplying an MD signal (typically sinusoidal and referred to as the carrier signal) with another signal that carries some information or message. In the frequency domain, the signal is moved from one frequency to another.
if then Typically the carrier signal is a sinusoidal signal and in various applications. The figures below illustrate a quick example of a 2-D modulation. The original signal from (3) is modulated with a sinusoidal signal to get (4). The equations (5) and (6) are the real and the imaginary components of the modulated signal.
Background/Motivation The MD modulation is one of the properties of the Multidimensional Fourier Transform.
MD Fourier Transform (FT) Fourier Transform (FT) of multi-dimensional (MD) signal or system is the transform of the MD signal or system that decomposes it into its frequency components. Essentially, it is the frequency response of the MD signal or system, so it depicts the frequency characteristics of the signal or system. A special case of the MD transform is the 1-D Fourier transform.
Computation The formula for computing the Fourier transform of an MD signal is
X ( ω 1 , ω 2 , . . . , ω M ) = ∑ n 1 = − ∞ ∞ ∑ n 2 = − ∞ ∞ . . . ∑ n M = − ∞ ∞ x ( n 1 , n 2 , . . . , n M ) e − j ( ω 1 n 1 + ω 2 n 2 + , . . . , + ω M n M ) {\displaystyle X(\omega _{1},\omega _{2},...,\omega _{M}){=}\sum _{n_{1}=-\infty }^{\infty }\sum _{n_{2}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }x(n_{1},n_{2},...,n_{M})e^{-j(\omega _{1}n_{1}+\omega _{2}n_{2}+,...,+\omega _{M}n_{M})}}
If the frequency response is given instead, then the inverse FT formula is used to derive the input signal or system. In this case the formula used is:
x ( n 1 , n 2 , . . . , n M ) = 1 ( 2 π ) M ∫ − π π ∫ − π π . . . ∫ − π π X ( ω 1 , ω 2 , . . . , ω M ) e j ( ω 1 n 1 + ω 2 n 2 + , . . . , + ω M n M ) d ω 1 d ω 2 . . . d ω M {\displaystyle x(n_{1},n_{2},...,n_{M}){=}{\frac {1}{(2\pi )^{M}}}\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }...\int \limits _{-\pi }^{\pi }X(\omega _{1},\omega _{2},...,\omega _{M})e^{j(\omega _{1}n_{1}+\omega _{2}n_{2}+,...,+\omega _{M}n_{M})}d\omega _{1}d\omega _{2}...d\omega _{M}}
Approaches / Extended Applications of MD Modulation
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